In the broadest terms, momentum conservation dictates that in order for an on-board effect to accelerate the spacecraft, the spacecraft must eject mass or emit radiation. As no significant anomaly occurred in the Pioneer 10 and 11 missions, it is unlikely that either spacecraft lost a major component during their cruise. In any case, such an occurrence would have resulted in a one time change in the spacecraft’s velocity, not any long-term acceleration. Therefore, it is safe to consider only the emission of volatiles as a means of mass ejection. Such emissions can be intentional (as during maneuvers) or due to unintended leaks of propellant or other volatiles on board. Radiation emitted as radiative energy is produced by on-board processes.

The spin of the Pioneer spacecraft makes it possible to apply a simplified treatment of forces of on-board origin that change slowly with time. Let us denote the unit vector normal to the spacecraft’s plane of rotation (i.e., the spin axis, which we assume to remain constant in time) by . Then, considering a force that is a linear function of time in a co-rotating reference frame that is attached to the spacecraft, it can be described in a co-moving (nonrotating) inertial frame as

where is the component of parallel with the spin axis and is the perpendicular component of . The component accelerates the spacecraft in the spin axis direction. The displacement due to the perpendicular component can be obtained by double integration with integration limits of and : describing an arithmetic spiral around the spin axis. This spiral vanishes (i.e., the displacement of the spacecraft remains confined along the spin axis) if , and even for nonzero its radius increases only linearly with time, and thus in most cases, it can be ignored safely.

There were several hundred^{20}
Pioneer 10 and Pioneer 11 maneuvers during their entire missions. The modeling of maneuvers entails
significant uncertainty due to several reasons. First, the duration of a thruster firing is known only
approximately and may vary between maneuvers due to thermal and mechanical conditions, aging,
manufacturing deficiencies in the thruster assembly, and other factors. Second, the thrust can vary as a
result of changing fuel temperature and pressure. Third, imperfections in the mechanical mounting of a
thruster introduce uncertainties in the thrust direction. Lastly, after a thruster has fired, leakage may occur,
producing an additional, small amount of slowly decaying thrust. When combined, these effects result in a
velocity change of several mm/s.

By the time Pioneer 11 reached Saturn, the behavior of its thrusters was believed to be well understood [27]. The effectively instantaneous velocity change caused by the firing of a thruster was followed by several days of decaying acceleration due to gas leakage. This acceleration was large enough to be observable in the Doppler data [270].

The Jet Propulsion Laboratory’s analysis of Pioneer orbits included either an instantaneous velocity increment at the beginning of each maneuver (instantaneous burn model) or a constant acceleration over the duration of the maneuver (finite burn model) [27]. In both cases, the burn is characterized by a single unknown parameter. The gas leak following the burn was modeled by fitting to the post-maneuver residuals a two-parameter exponential model in the form of

with and being the unknown parameters. The typical magnitude of is several mm/s, while the time constant is of order 10 days. Due to the spin of the spacecraft, only acceleration in the direction of the spin axis needs to be accounted for, as accelerations perpendicular to the spin axis are averaged out over several resolutions [27, 380].

Regardless of the source of a leak, the effects of outgassing on the spacecraft are governed by the rocket equation [27]:

where the dot denotes differentiation with respect to time, and is the exhaust velocity. The Pioneer spacecraft mass is approximately . For comparison, the anomalous acceleration, , requires an outgassing rate of 6.89 g/yr at an exhaust velocity of 1 km/s. The exhaust velocity of a hot gas, according to the rocket engine nozzle equation, can be calculated
as [367]^{21}:

A review of the Pioneer 10 and 11 spacecraft design reveals only three possible sources of outgassing: the propulsion system (fuel leaks), the radioisotope thermoelectric generators, and the battery.

The propulsion system carried 30 kg of hydrazine propellant and N pressurant. Loss of either due to a leak could produce a constant or slowly changing acceleration term. Propellant and pressurant can be lost due to a malfunction in the propulsion system, and also due to the regular operation of thruster valves, which are known to have small, persistent leaks lasting days or even weeks after each thruster firing event, as described above in Section 4.4.1. While the possibility of additional propellant leaks cannot be ruled out, in order for such leaks to be responsible for a constant acceleration like the anomalous acceleration of Pioneer 10 and 11, they would have had to be i) constant in time; ii) the same on both spacecraft; iii) not inducing any detectable changes in the spin rate or precession. Given these considerations, Anderson et al. conservatively estimate that undetected gas leaks introduce an uncertainty not greater than

Outgassing can also occur in the radioisotope thermoelectric generators as a
result of alpha decay. Each kg of ^{238}Pu produces 0.132 g of helium annually;
the total amount of helium produced by the approx. 4.6 kg of radioisotope fuel on
board^{22}
is, therefore, 0.6 g/year. Exterior temperatures of the RTGs at no point exceeded 320 ° F=433 K.
According to Equation (4.21), the corresponding exhaust velocity is 2.13 km/s, resulting in an acceleration
of . (This is slightly larger than the corresponding estimate in [27], where the authors
adopted the figures of and .) However, the circumstances required
to achieve this acceleration are highly unrealistic, requiring all the helium to be expelled at
maximum efficiency and in the spin axis direction. Using a more realistic (but still conservative)
scenario, Anderson et al. estimate the bias and error in acceleration due to He-outgassing as

Another source of possible outgassing not previously considered may be the spacecraft’s battery.
According to Equation (4.21), H_{2} gas leaving the battery system at a temperature of 300 K can acquire an
exhaust velocity of 92.6 m/s. For O_{2} at 300 K, the exhaust velocity is 23.2 m/s. At these velocities, an
outgassing of 74 g/year of H_{2} or 298 g/year of O_{2} can produce an acceleration equal to , so the
battery cannot be ruled out in principle as a source of a near constant acceleration term. However, no
realistic construction [83] for a 5 A, 11.3 V AgCd battery would provide near enough volatile electrolites
for such outgassing to occur, and in any case, the nominal performance of the battery system for a far
longer time period than designed indicates that no significant loss of volatiles from the battery has
taken place. A conservative (but still generous) estimate using a battery of maximum weight,
2.35 kg, assuming a loss of 10% of its mass over 30 years, and a thrust efficiency of 50% yields

The spacecraft carried several on-board energy sources that produced waste heat (see Section 2.4). Most notably among these are the RTGs; additional heat was produced by electrical instrumentation. Further heat sources include Radioisotope Heater Units and the propulsion system.

As the spacecraft is in an approximate thermal steady state, heat generated on board must be removed from the spacecraft [380]. In deep space, the only mechanism of heat removal is thermal radiation: the spacecraft can be said to be radiatively coupled to the cosmic background, which can be modeled by surrounding the spacecraft with a large, hollow spherical black body at the temperature of 2.7 K.

As the spacecraft emits heat in the form of thermal photons, these also carry momentum , in accordance with the well known law of , where is the photon’s frequency, is Planck’s constant, and is the velocity of light. This results in a recoil force in the direction opposite to that of the path of the photon. For a spherically symmetric body, the net recoil force is zero. However, if the pattern of radiation is not symmetrical, the resulting anisotropy in the radiation pattern yields a net recoil force.

The magnitude of this recoil force is a subject of many factors, including the location and thermal power of heat sources, the geometry, physical configuration, and thermal properties of the spacecraft’s materials, and the radiometric properties of its external (radiating) surfaces.

Key questions concerning the thermal recoil force that have been raised during the study of the Pioneer anomaly include [164, 245, 327]:

- How much heat from the RTGs is reflected by the spacecraft, notably the rear of its high-gain antenna, and in what direction?
- Was there a fore-aft asymmetry in the radiation pattern of the RTGs due to differential aging?
- How much electrical heat generated on-board was radiated through the spacecraft’s louver system?

The recoil force due to on-board generated heat that was emitted anisotropically was recognized early as a possible origin of the Pioneer anomaly. The total thermal inventory on board the Pioneer spacecraft exceeded 2 kW throughout most of their mission durations. The spacecraft were in an approximate steady state: the amount of heat generated on-board was equal to the amount of heat radiated by the spacecraft.

The mass of the Pioneer spacecraft was 250 kg. An acceleration of is equivalent to a force of acting on a 250 kg object. This is the amount of recoil force produced by a 65 W collimated beam of photons. In comparison with the available thermal inventory of 2500 W, a fore-aft anisotropy of less than 3% can account for the anomalous acceleration in its entirety. Given the complex shape of the Pioneer spacecraft, it is certainly conceivable that an anisotropy of this magnitude is present in the spacecrafts’ thermal radiation pattern.

The issue of the thermal recoil force remains a subject of on-going study, as estimates of the actual magnitude of this force may require significant revision in the light of new data and new investigations [378, 379, 380, 397].

Throughout most of their missions, the Pioneer 10 and 11 spacecraft were transmitting continuously in the direction of the Earth using a highly focused microwave radio beam that was emitted by the high gain antenna (HGA; see Section 2.4.5). The recoil force due to the radio beam can be readily calculated.

A naive calculation uses the nominal value of the radio beam’s power (8 W), multiplied by the reciprocal of the velocity of light, , to obtain the radio beam recoil force. This is a useful way to estimate the recoil force, but it may need refinement.

The spacecraft’s radio transmission is concentrated into a very narrow beam: signal attenuation exceeds 20 dB at only 3.75° deviation from the antenna centerline (see Figure 3.6-13 in [292]). The projected transmitter power in the beam direction can be computed using the integral where is the angular power distribution of the antenna. Given the antenna power distribution, we find that , where is the total power radiated by the antenna, to an accuracy much better than 1%. For this reason, the shape of the transmission beam needs not be taken into account when computing the recoil force.

However, as discussed in Section 2.4.5, the power of the spacecraft’s transmitter was not constant in time: if the telemetry readings are accepted as reliable, transmitter power may have decreased by as much as 3 W or more near the end of Pioneer 10’s mission. Furthermore, some (estimated 10%) of the radio beam may have missed the antenna dish altogether, resulting in a reduced efficiency with which the energy of the spacecraft’s transmitter is converted into momentum.

Note that the navigational model that was used to navigate the Pioneers did not include this effect. It became clear only recently leading to the need to include this model as a part of the on-going efforts to re-analyze the Pioneer data.

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